Question: Solve for $x$ : $ 3|x - 8| + 6 = 4|x - 8| + 4 $
Explanation: Subtract $ {3|x - 8|} $ from both sides: $ \begin{eqnarray} 3|x - 8| + 6 &=& 4|x - 8| + 4 \\ \\ {- 3|x - 8|} && {- 3|x - 8|} \\ \\ 6 &=& 1|x - 8| + 4 \end{eqnarray} $ Subtract $4$ from both sides: $ \begin{eqnarray} 6 &=& 1|x - 8| + 4 \\ \\ {- 4} && {- 4} \\ \\ 2 &=& 1|x - 8| \end{eqnarray} $ Simplify: $ 2 = |x - 8| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -2 = x - 8 $ or $ 2 = x - 8 $ Solve for the solution where $x - 8$ is negative: $ - 2 = x - 8$ Add ${8}$ to both sides: $ \begin{eqnarray} - 2 &=& x - 8 \\ \\ {+ 8} && {+ 8} \\ \\ -2 + 8 &=& x \end{eqnarray} $ $ 6 = x $ Then calculate the solution where $x - 8$ is positive: $ 2 = x - 8 $ Add ${8}$ to both sides: $ \begin{eqnarray} 2 &=& x - 8 \\ \\ {+ 8} && {+ 8} \\ \\ 2 + 8 &=& x \end{eqnarray} $ $ 10 = x $ Thus, the correct answer is $x = 6 $ or $x = 10 $.